Unweighted Index Numbers
There are two methods of constructing unweighted index numbers: (1) Simple Aggregative Method (2) Simple Average of Relative Method
Simple Aggregative Method
In this method, the total price of commodities in a given (current) year is divided by the total price of commodities in a base year and expressed as percentage:
\[{P_{on}} = \frac{{\sum {P_n}}}{{\sum {P_o}}} \times 100\]
Simple Average of Relative Method
In this method, we compute price relatives or link relatives of the given commodities and then use one of the averages such as the arithmetic mean, geometric mean, median, etc. If we use the arithmetic mean as the average, then:
\[{P_{on}} = \frac{1}{n}\sum \left( {\frac{{{P_n}}}{{{P_o}}}} \right) \times 100\]
The simple average of relative method is simpler and easier to apply than the simple aggregative method. The only disadvantage is that it gives equal weight to all items.
Example:
The following are the prices of four different commodities for $$1990$$ and$$1991$$. Compute a price index with the (1) simple aggregative method and (2) average of price relative method by using both the arithmetic mean and geometric mean, taking $$1990$$ as the base.
|
Commodity
|
Cotton
|
Wheat
|
Rice
|
Grams
|
|
Price in $$1990$$
|
$$909$$
|
$$288$$
|
$$767$$
|
$$659$$
|
|
Price in $$1991$$
|
$$874$$
|
$$305$$
|
$$910$$
|
$$573$$
|
Solution:
The necessary calculations are given below:
|
Commodity
|
Price in $$1990$$
$${P_o}$$ |
Price in $$1991$$
$${P_n}$$ |
Price Relative
$$P = \frac{{{P_n}}}{{{P_o}}} \times 100$$ |
$$\log P$$
|
|
Cotton
|
$$909$$
|
$$874$$
|
$$\frac{{874}}{{909}} \times 100 = 69.15$$
|
$$1.9829$$
|
|
Wheat
|
$$288$$
|
$$305$$
|
$$\frac{{305}}{{288}} \times 100 = 105.90$$
|
$$2.0249$$
|
|
Rice
|
$$767$$
|
$$910$$
|
$$\frac{{910}}{{767}} \times 100 = 118.64$$
|
$$2.0742$$
|
|
Grams
|
$$659$$
|
$$573$$
|
$$\frac{{573}}{{659}} \times 100 = 86.95$$
|
$$1.9393$$
|
|
Total
|
$$\sum {P_o} = 2623$$
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$$\sum {P_n} = 2662$$
|
$$\sum P = 407.64$$
|
$$\begin{gathered} \sum \log P \\ = 8.0213\\ \end{gathered} $$
|
(1) Simple Aggregative Method:
\[{P_{on}} = \frac{{\sum {P_n}}}{{\sum {P_o}}} \times 100 = \frac{{2662}}{{2623}} \times 100 = 101.49\]
(2) Average of Price Relative Method (using the arithmetic mean):
\[{P_{on}} = \frac{1}{n}\sum \left( {\frac{{{P_n}}}{{{P_o}}}} \right) \times 100 = \frac{1}{4}\left( {407.64} \right) = 101.91\]
Average of Price Relative Method (using the geometric mean):
\[{P_{on}} = anti\log \left( {\frac{{\sum \log P}}{n}} \right) = anti\log \left( {\frac{{8.0213}}{4}} \right) = 101.23\]